# Direct product of A5 and Z2

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## Definition

This group is defined in the following equivalent ways:

1. It is the full icosahedral group: it is the group of all rigid symetries of the regular icosahedron, including both orientation-preserving symmetries and orientation-reversing symmetries.
2. It is the external direct product of the alternating group of degree five and the cyclic group of order two.

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 120#Arithmetic functions

### Basic arithmetic functions

Function Value Similar groups Explanation for function value
order (number of elements, equivalently, cardinality or size of underlying set) 120 groups with same order order of direct product is product of orders, so the order is $|A_5||\mathbb{Z}_2| = (5!/2)(2) = 5! = 120$
exponent of a group 30 groups with same order and exponent of a group | groups with same exponent of a group exponent of direct product is lcm of exponents, so lcm of exponents of $A_5$ and $\mathbb{Z}_2$, which is $\operatorname{lcm} \{ 30, 2 \} =30$
composition length 2 groups with same order and composition length | groups with same composition length direct product of two simple groups
chief length 2 groups with same order and chief length | groups with same chief length direct product of two simple groups
max-length 5 groups with same order and max-length | groups with same max-length
Frattini length 1 groups with same order and Frattini length | groups with same Frattini length
nilpotency class not a nilpotent group
derived length not a solvable group
Fitting length not a solvable group

## Group properties

Property Satisfied? Explanation
abelian group No
nilpotent group No
solvable group No
simple group, simple non-abelian group No
almost simple group No
quasisimple group No
almost quasisimple group No
semisimple group No
perfect group No
directly indecomposable group No

## GAP implementation

### Group ID

This finite group has order 120 and has ID 35 among the groups of order 120 in GAP's SmallGroup library. For context, there are 47 groups of order 120. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(120,35)

For instance, we can use the following assignment in GAP to create the group and name it $G$:

gap> G := SmallGroup(120,35);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [120,35]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.

### Other descriptions

Description Functions used
DirectProduct(AlternatingGroup(5),CyclicGroup(2)) DirectProduct, AlternatingGroup, CyclicGroup